Residually Finite Groups
Our object in this chapter is to discover groups which are residually finite: in particular we shall find that free groups and free products of residually finite groups are residually finite and that several significant types of soluble groups share this
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Herausgegeben von P.R. Halmos • P.J. Hilton R. Remmert· B. Szokefalvi-Nagy Unter Mitwirkung von L.V.Ahlfors • R. Baer F. L. Bauer • A. Dold •]. L. Doob S. Eilenberg • M. Kneser • G. H. Miiller M. M. Postnikov • B. Segre • E. Sperner Geschaftsfiihrender Herausgeber: P. J. Hilton
Derek J. S. Robinso n
Finiteness Conditions and Generalized Soluble Group s Part 2
Springer-Verlag Berlin Heidelberg GmbH
1972
Derek
J. S. Robinson
University of Illinois Urbana, in.
AMS Subject Classifications (1970): Primary 20E15, 20E25, 20E99 Secondary 20F30, 20F35, 20F45, 20F50, 20F99
ISBN 978-3-642-05712-0 ISBN 978-3-662-11747-7 (eBook) DOI 10.1007/978-3-662-11747-7
This work is subject to copyright. All rights are reserved, whether the whole Qr part of the material is concerned, specifically those of translation reprinting re~use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under §54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
@ by Springer-Verlag Berlin Heidelberg 1972. Originally published by Springer-Verlag Berlin Heidelberg New York in 1972 Softcover reprint of the hardcover 1st edition 1972 Library of Congress Catalog Card Number 74-189458.
Preface
This book is a study of group theoretical properties of two disparate kinds, firstly finiteness conditions or generalizations of finiteness and secondly generalizations of solubility or nilpotence. It will be particularly interesting to discuss groups which possess properties of both types. The origins of the subject may be traced back to the nineteen twenties and thirties and are associated with the names of R. Baer, S. N. Cernikov, K. A. Hirsch, A. G. Kuros, 0. J. Schmidt and H. Wielandt. Since this early period, the body of theory has expanded at an increasingly rapid rate through the efforts of many group theorists, particularly in Germany, Great Britain and the Soviet Union. Some of the highest points attained can, perhaps, be found in the work of P. Hall and A. I. Mal'cev on infinite soluble groups. Kuras's well-known book "The theory of groups" has exercised a strong influence on the development of the theory of infinite groups: this is particularly true of the second edition in its English translation of 1955. To cope with the enormous increase in knowledge since that date, a third volume, containing a survey of the contents of a very large number of papers but without proofs, was added to the book in 1967. Despite this useful addition and the books of M. Hall, E. Schenkman and W. R. Scott, which deal with finite as well as infinite groups, there is a clear need for a detailed account of the theory of finiteness conditions and of generalized soluble and nilpotent groups. The present work represents an attempt to meet this need. I have sought to collect the most important results in the theory, which are scattered throughout the literature, and to present t
